Unique continuation and Hardy's uncertainty principle for hyperbolic Schr\"odinger equations

We prove unique continuation properties related to the Hardy uncertainty principle for solutions of the hyperbolic nonlinear Schr\"odinger equation and the hyperbolic Schr\"odinger equation with potential. Under suitable conditions on the nonlinearity, or the potential, we show that if $u$ is a solution with Gaussian decay at two different times, then $u\equiv 0$. These results extend to the hyperbolic setting the work of Escauriaza, Kenig, Ponce, and Vega (JEMS, 10, 2008) for the classical Schr\"odinger equation. The proofs rely on Carleman estimates based on calculus and convexity arguments, with the main challenge being to provide a rigorous justification of these estimates. Although our approach follows the general strategy of Escauriaza, Kenig, Ponce, and Vega, several technical modifications are required to handle the hyperbolic character of the equation.